Exponential Equations

Introduction to Exponential Functions

100

What is the definition of an exponential function?

An exponential function is a mathematical function in which the variable appears in the exponent

200

§Write the general notation for an exponential function.

The general notation for an exponential function is f(x) = a * b^x, where a is the initial value, b is the base, and x is the input variable.

300

How are exponential functions represented on a graph?

 Exponential functions are represented on a graph as a curve that either increases (exponential growth) or decreases (exponential decay) as x increases.

400

Name two properties of exponential functions. (Hint:Think about Domain and Range)

1) The domain is all real numbers. 2) The range is either positive or negative, depending on

500

216 × 36 ²ⁿ = ( 1 /216 )^-n

-3

Solving Exponential Equations

100

What are the basic techniques for solving exponential equations?

The basic techniques for solving exponential equations are making the base same and using properties of exponents, and simplifying the equation to isolate the variable.

200

3^x+2 = 1/9

x = -4

300

Solve the exponential equation: 3^(2x) = 27.

x=3/2

400

Solve the exponential equation: 2^(x + 1) = 8.

x=2

500

Give an example of an application problem that involves solving exponential equations.

Determining the time it takes for a radioactive substance to decay to a certain level given its half-life.

Exponential Growth and Decay

100

Define exponential growth and provide an example.

Exponential growth is a type of growth in which a quantity increases at an accelerating rate proportional to its current value. Example: Population growth in an idealized environment where resources are unlimited.

200

27^X = 9^X-2

X= -1/4

300

8^3X+1 = 4^2X

-3/5

400

16 × 64^3-2r = ( 1 /4 ) ^3r

11/3

500

9^1-3k × 243^2k = 27^3k

2/5

Exponential Growth and Decay Problems

100

2^3X = 2^3-X

0.5

200

6^K+1 = 216

K =2

300

(1/5)^-3n = 625

4/3

400

9^b× 81^-2b= 1

0

500

1/25 ×( 1 /25 )-n = 25 ^-3n

3/8

Find Domain and Range

100

Find the Domain and Range of f(x)=5^(x)

Domain: (−∞,∞)

Range: (0,∞)

200

Find the Domain and Range of f(x)=6^(x) -5

Domain: (−∞,∞)

Range: (−5,∞)

300

Find the Domain and Range of f(x)=-2^(x) +1

Domain: (−∞,∞)

Range: (−∞,1)

400

Find the Domain and Range of f(x)=-3^(x) -5

Domain: (−∞,∞)

Range: (−∞,-5)

500

Find the Domain and Range f(x)=-3^(x)+5 

Domain: (−∞,∞)

Range: (−∞,5)